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HCF and LCM

Arithmetic

Factors and Multiples

If a number a divide b exactly, then a is called a factor of b and b is called the multiple of a.

Example:

64 is a number.
It is divisible by 2,4,8,16 and 32 exactly. So, 2,4,8,16,32 are its factors and 64 is a multiple of all.

Prime Factor

When the factor of a number is a prime number, then it is called a prime factor.

Example:

105 is a number.
It is divisible by 3,5 and 7exactly. As all are prime numbers therefore they are the prime factors.

Common Factors

When two or more numbers has the same number as a factor it is called a common factor of all the numbers.

Example:

105,35,15 are numbers.
105 has 3, 5 and 7 as prime factors. 35 has 7 and 5 as prime factors. 15 has 5 and 3 as prime factors. 5 is common in all hence it is a common factor.

Highest Common Factor (H.C.F.) or Greatest Common Divisor (G.C.D.)

The H.C.F. or G.C.D. of two or more numbers is the largest number that divides each of them exactly.

Example:

105 and 70 are numbers.
105 has 3, 5 and 7 as prime factors. 70 has 2, 5 and 7 as prime factors. 5 and 7 are common to both. Hence the product of 5 and 7 i.e. 35 is the H.C.F. or G.C.D.

Prime factorization method for finding H.C.F.

  • Step 1: Express each of the given numbers as the product of prime factors.
  • Step 2: The product of terms containing least powers of common prime factors gives the H.C.F. of the given numbers.

Long Division method for finding H.C.F.

  • Step 1: Divide the larger number by the smaller number and obtain the remainder.
  • Step 2: Divide the divisor by the remainder obtained in Step 1.
  • Step 3: Repeat the process of dividing the preceding divisor by the remainder last obtained till you get 0 as remainder.
  • Step 4: The last divisor is the required H.C.F.

Least common multiple (L.C.M.)

L.C.M. of two or more numbers is the least number which can be divided exactly by each of the given numbers.

Example:

15 and 20 are numbers.
15 has 3 and 5 as factors. 20 has 5 and 4 as factors. The product of all the factors 3, 5 and 4 is 60. This is L.C.M. of the numbers.

Prime factorization method for finding L.C.M.

  • Step 1: Express each of the given numbers as the product of prime factors.
  • Step 2: The product of terms containing highest powers of all the factors gives the L.C.M. of the given numbers.

Common Division method for finding L.C.M.

  • Step 1: Arrange the given number in a row.
  • Step 2: Divide by a number which divides exactly at least two of the given numbers and carry forward the numbers which are not divisible.
  • Step 3: Repeat the above process till no two numbers are divisible by the same number except 1.
  • Step 4: The product of the divisors and the undivided numbers is the required L.C.M. of the given numbers.

Relationship between numbers, their H.C.F and L.C.M.

Product of two given numbers = Product of their H.C.F. and L.C.M.
15 and 20 are numbers.
H.C.F. is 5 and L.C.M. is 60. Product of numbers = L.C.M. × H.C.F. = 60 × 5 = 300.

H.C.F and L.C.M. of fractions

H.C.F. of given fractions = H.C.F. of numerators/L.C.M. of denominators
L.C.M. of given fractions = L.C.M. of numerators/H.C.F. of denominators

Factors

2a2bc has 2,a,b and c as factors. Factors are used to find the L.C.M. and H.C.F. of expressions. L.C.M. and H.C.F. also play an important role in mathematics. Suppose you are given some sticks of different sizes and you are asked to find a length whose length will be divisible by all the sticks of different sizes. You need the knowledge of L.C.M. Again consider you are given many sticks and you are asked to find a length which will measure all of them in integer terms. You will need the knowledge of H.C.F.

L.C.M.

Rules to find L.C.M. (Lowest common multiple)

  1. Find the L.C.M. of the constant terms.
  2. Find the power of all the expressions for each variable or symbolic constant.
  3. Find the maximum power for each variable or symbolic constant among each expression.
  4. List the variables with the maximum power of each.
  5. Multiply it with the L.C.M. of the constant term.

Evaluate L.C.M. of 3a2b6c and 8a3b2cd
  1. L.C.M. of the constant terms is 24.
  2. Power of all the expressions for each variable or symbolic constant.
    a:2 and 3
    b:6 and 2
    c:1 and 1
    d:0 and 1
  3. Maximum power for each variable or symbolic constant among each expression.
    a:2 and 3 max = 3
    b:6 and 2 max = 6
    c:1 and 1 max = 1
    d:0 and 1 max = 1
  4. List the variables with the maximum power of each.
    a3b6cd
  5. Multiply it with the L.C.M. of the constant term.
    24a3b6cd
L.C.M. is 24a3b6cd

H.C.F. or G.C.D.

Rules to find G.C.D. or H.C.F. (Greatest common Divisor or Highest Common Factor)

  1. Find the G.C.D. of the constant terms.
  2. Find the power of all the expressions for each variable or symbolic constant.
  3. Find the minimum power for each variable or symbolic constant among each expression.
  4. List the variables with the minimum power of each.
  5. Multiply it with the G.C.D. of the constant term.

Evaluate G.C.D. of 3a2b6c and 8a3b2cd
  1. G.C.D. of the constant terms is 1.
  2. Power of all the expressions for each variable or symbolic constant.
    a:2 and 3
    b:6 and 2
    c:1 and 1
    d:0 and 1
  3. Minimum power for each variable or symbolic constant among each expression.
    a:2 and 3 max = 2
    b:6 and 2 max = 2
    c:1 and 1 max = 1
    d:0 and 1 max = 0
  4. List the variables with the maximum power of each.
    a2b2cd0
  5. Multiply it with the G.C.D. of the constant term.
    a2b2c
G.C.D. is a2b2c



Algebra

H.C.F. of Monomials

The H.C.F. of two or more monomials is the product of the H.C.F. of their numerical coefficients and each common literal raised to the lowest power.

Example:

3a3b2c and 6a2bc2 has H.C.F. as 3a2bc.

L.C.M. of Monomials

The L.C.M. of two or more monomials is the product of the L.C.M. of their numerical coefficients and each literal raised to the highest power.

Example:

3a3b2c and 6a2bc2 has L.C.M. as 6a3b2c2.

H.C.F. of Polynomials

The H.C.F. of two or more polynomials is the product of the H.C.F. of their numerical coefficients and each common factor raised to the lowest power.

L.C.M. of Polynomials

The L.C.M. of two or more polynomials is the product of the L.C.M. of their numerical coefficients and each factor raised to the highest power.

Example:

a2 - b2 and a2 + ab
Factors of a2 - b2 = (a + b)(a - b)
Factors of a2 + ab = a(a + b)
H.C.F. = (a + b)
L.C.M. = a(a + b)(a - b)


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